the current normal to its plane is so that the quantity which enters through this triangle is
The quantities which enter through the triangles and respectively are
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If is the component of the velocity in the direction then the quantity which leaves the tetrahedron through is
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Since this is equal to the quantity which enters through the three other triangles,
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multiplying by we get
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| (1) |
If we put |
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and make such that
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then |
| (2) |
Hence, if we define the resultant current as a vector whose magnitude is and whose direction-cosines are and if denotes the current resolved in a direction making an angle with that of the resultant current, then
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| (3) |
shewing that the law of resolution of currents is the same as that of velocities, forces, and all other vectors.
287.] To determine the condition that a given surface may be a surface of flow.
Let |
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be the equation of a family of surfaces any one of which is given by making constant, then, if we make
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| (5) |
the direction-cosines of the normal, reckoned in the direction in which increases, are
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| (6) |